# Virtual representation motives

## TL;DR

This paper explores the motives of representation schemes of algebras, focusing on the challenges posed by non-trivial principal PGL_n-bundles and using Brauer-Severi schemes to address these in specific superpotential algebra cases.

## Contribution

It introduces a method to compute motives of representation schemes via Brauer-Severi schemes for certain superpotential algebras, advancing understanding in non-commutative algebraic geometry.

## Key findings

- Motives of certain superpotential algebra representation schemes can be computed using Brauer-Severi schemes.
- Principal PGL_n-bundles pose challenges in motive calculations, but special cases allow inductive approaches.
- The approach provides new insights into non-commutative algebraic geometry and motives.

## Abstract

Principal $GL_n$-bundles (aka vector bundles) are locally trivial in the Zariski topology, whereas principal $PGL_n$-bundles (aka Azumaya algebras) are not, to the delight of every non-commutative algebraist. Still, this makes the calculation of motives of representation schemes of algebras next to impossible. In very special cases, Brauer-Severi schemes (and their motives) can be used to tackle this problem inductively. We illustrate this in the case of certain superpotential algebras.